Normalization of correlated random variables in structural reliability analysis using fourth-moment transformation

In this paper, a fourth-moment transformation technique is proposed to transform correlated nonnormal random variables into independent standard normal ones. The procedure mainly includes two steps: First, the correlated nonnormal random variables are transformed into correlated standard normal ones using the fourth-moment transformation, where the complete mathematical formula of the correlation coefficient in standard normal space, i.e., equivalent correlation coefficient, is proposed and the upper and lower bounds of original correlation coefficient are identified to ensure the transformation executable; Second, the correlated standard normal random variables are transformed into independent standard normal ones using Cholesky decomposition. For the cases of original correlation matrix with very small eigenvalues, the equivalent correlation matrix might become a nonpositive semidefinite matrix. A recently developed method for solving the problem is adopted to make Cholesky decomposition ready. A first-order reliability method (FORM) for structural reliability analysis involving correlated random variables is developed using the proposed transformation technique. Several numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method for structural reliability assessment considering correlated random variables.